What's the difference between 'just' and 'fair'?

Because there's a lot of overlap between the two words, in many contexts they're interchangeable. But in some specific idiomatic forms (i.e. - "It's a fair cop", "We fight a just war") only one is used.

To the extent that there's a semantic difference, I would say that a fair settlement, for example, is one where the parties directly involved feel a satisfactory compromise has been reached. But to me at least, a just settlement is one that meets external criteria of correctitude (opinions of outsiders, formal moral/legal codes, etc.).

As a teenager, I attended an "Institute in Basic Youth Conflicts" seminar, where — among other things — I read a pamphlet entitled "Fairness: the Unexpected Enemy of Justice". The upshot was that justice comes from God, but fairness is a human construct. I thought that was a bunch of hooey then, and I still do — but the concept has some merit.

A useful distinction is that justice is objective, while fairness is subjective. A judge's sentence may be just, because it is based on a law that is the same for all, and yet not seem fair because it fails to take circumstances into account.

Even this is a bit slippery, because standards of justice vary widely throughout and between societies — e.g. cutting off a thief's hand would be considered justice in Saudi Arabia but a heinous crime in the United States — but I think it generally holds true.

I agree with Jim's comment; however, The American Heritage Dictionary, 4th edition, had this interesting note under the entry for fair:

Synonyms fair, just, equitable, impartial, unprejudiced, unbiased, objective, dispassionate These adjectives mean free from favoritism, self-interest, or preference in judgment. Fair is the most general: a fair referee; a fair deal. Just stresses conformity with what is legally or ethically right or proper: "a just and lasting peace" (Abraham Lincoln)...

"Just" refers to an action justified under the circumstances. "Fair" refers to an action that treats people as they deserve to be treated. Many times, actions that are just are not fair. In hard cases, an action may be justified because there aren't superior options, even if it's results are unfair to at least some people.

If a madman holding a single hostage is going to blow up a school full of children, shooting him through the hostage may be just, but it isn't fair to the hostage.

In addition, outcomes that aren't the results of human action are neither just nor unjust. For example, a hurricane is neither just nor unjust. Yet a hurricane can be very unfair. One lazy person wins the lottery, another more deserving person does not. There's nothing unjust about that, but it's not fair.